Classes of finite groups with generalized subnormal cyclic primary subgroups

We study the properties of the classes νπℌ (νπ*ℌ) of finite groups whose all cyclic primary π-subgroups are ℌ-subnormal (respectively, K-ℌ-subnormal) for a set of primes π and a hereditary homomorph ℌ. It is established that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\pi \mathfrak{F}$$\end{document} is a hereditary saturated formation if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{F}$$\end{document} is a hereditary saturated formation. We in particular obtain some new criteria for the p-nilpotency and ϕ-dispersivity of finite groups. A characterization of formations with Shemetkov property is obtained in the class of all finite soluble groups.